Journal of Hydrology, 115 (1990) 231-241 231 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

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R E L A T I V E P E R F O R M A N C E O F A S O I L M O I S T U R E A C C O U N T I N G M O D E L I N E S T I M A T I N G R E T U R N F L O W

K.M. MOHAN RAO 1, DEEPAK KASHYAP 1 and SATISH CHANDRA 2

~Department of Hydrology, University of Roorkee, Roorkee (India) 2National Institute of Hydrology, Roorkee (India)

(Received October 22, 1988; accepted after revision August 30, 1989)

ABSTRACT

Mohan Rao, K.M., Kashyap, D. and Satish Chandra, 1990. Relative performance of a soil moisture accounting model in estimating return flow. J. Hydrol., 115: 231-241.

The current practice of estimating return flow (also known as ground water recharge) from irrigation and rainfall is mostly based on soil moisture accounting (SMA) models and involves many restrictive assumptions. Prominent among them is the existence of a threshold moisture content termed ~field capacity', which must be quantified as a flow parameter to be compatible with its hydraulic implication in the SMA models. This paper presents (i) a method for quantifying field capacity as a flow parameter, and (ii) assessment of a proposed SMA model for estimation of return flow, in comparison with an earlier distributed model based on a finite difference solution of the Richards' equation, for studying the possibility of replacement of the more elaborate distributed model by the easy to adopt SMA model. The method for quantifying field capacity as a flow parameter is more suitable for coarse soils. The assumptions implicit in the SMA model result in estimated daily rates of return flow which are not acceptable although seasonal totals are reasonably acceptable.

INTRODUCTION

Rainfal l and i r r iga t ion wate r pass t h r o u g h the unsa tu r a t ed zone, before par t of it reaches the wate r table in the form of r e tu rn flow. Thus, the volume and t ime dis t r ibut ion of r e tu rn flow is governed by the uns t eady state flow of water in the unsa tu r a t ed zone. Current ly , s imula t ion of this flow process is most ly based on soil mois ture accoun t ing (SMA) models, in which the ent ire un- sa tu ra ted zone is considered as a unit. These models involve many res t r ic t ive assumpt ions re la t ing to the flow process. M a n y of these assumptions, par- t i cu la r ly the p rominen t concep tua l ent i ty 'field capaci ty ' , are re laxed in the dis t r ibuted models governed by the Richards ' equat ion. However , s implici ty of the SMA models makes them preferable to the dis t r ibuted models. Thus, if it is desired to adopt the SMA models in place of the dis t r ibuted models, two in teres t ing ques t ions will be: (i) the accep tance of the est imates of the re tu rn flow, with reference to the es t imates by the dis t r ibuted model; (ii) the possibil- i ty of quan t i fy ing field capac i ty as a flow parameter , so tha t its hydrau l i c impl icat ions in the SMA models are more realistic.

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232 K.M. MOHAN RAO ET AL.

REVIEW

Soil moisture accounting models involve a book keeping of various mass balance components of the unsaturated zone, taken as a unit (Satish Chandra, 1979). Hence, they require less expertise and computational effort than Richards' equation based (distributed) models. Usually, water movement in the root zone as defined by Bear (1972) is modelled and the rest of the unsaturated zone is assumed to be at field capacity at all times. Surface runoff is evaluated by some land use model (e.g. Victor, 1964). The excess infiltration (i.e. infil- tration-soil moisture deficit-actual evapotranspiration) is assumed to be available, instantaneously, as return flow. Richards (1960) proposed a moratorium on the concept of field capacity. Miller (1964) observed that the equivalent of 0.33 bar cannot reasonably provide a measure of field capacity. Sykes and Loomis (1967) and Mohan Rao et al. (1986) have stressed the need to quantify field capacity as a flow parameter. Miller (1967) and Rushton and Ward (1979) have reported under-estimation of return flow by SMA models. This could be because of the current state of the art of quantifying the field capacity and the associated restrictive assumptions.

PRESENT STUDY

In a previous study the authors (Mohan Rao et al., 1986) simulated soil moisture movement by a numerical solution of the Richards' equation. The return flow was estimated for a series of infiltrations generated for a given series of irrigation and rainfall. While the surface runoff was neglected, surface ponding was accounted for. The study was carried out for two sequential crop seasons of rice and wheat, grown on two soils loam and clay. In the present study, the same infiltration series is routed through the unsaturated zone by the proposed SMA model, with identical input data which are described below. The return flow series so generated are compared with the corresponding series generated by the distributed model. Thus, the differences between the estimates of return flow, as computed by both models, can be attributed to the different routing procedures and to the additional restrictive assumptions in the SMA model. The objective is to study the possibility of replacement of the more elaborate distributed model by the simple SMA model.

The earlier distributed model

The authors developed a one-dimensional (vertical) distributed numerical (computer) model to simulate the unsteady state flow of moisture in the un- saturated zone extending from ground surface down to the water table (Mohan Rao et al., 1986). Richards' equation, in its local form for non-hysteretic flow condition and in the head form (eqn. (1)) is the governing equation:

- K ~ - ~ ( - h + Z ) - E (1) at Yz

SOIL MOISTURE ACCOUNTING MODEL 233

where:

h = capillary suction head [L] Z = elevation above water table (i.e., positive upwards) [L] 0 = volumetric moisture content K = capillary conductivity (a function of O/h and Z) [L T 1] t = time IT] E = actual evapotranspiration (a function of O/h, Z and t) [T -1] C = specific moisture capacity = O0/~h [L-l].

The calculation of actual evapotranspiration is carried out by adopting an appropriate sub-model. The governing equation is solved by the Crank-Nicol- son finite difference scheme (Remson et al., 1971). The resulting system of non-linear simultaneous equations is linearized explicitly. The solution obtained is verified, at prescribed time intervals, by solving the system implicitly by Picard's i teration method described in Remson et al. (1971) and comparing the corresponding [h] (i.e. the vector of capillary suction head). The model can account for soil layering; apart from this, time variation of the following are also accounted for: (i) position of water table; (ii) vegetal root zone depth; (iii) rainfall and irrigation intensity; (iv) evapotranspiration rate.

The ground surface boundary condition is automatically identified and assigned. The output of the model comprises the following.

(1) Time and depth distribution of moisture content as well as capillary suction head.

(2) Time distribution of ponded depth of water, surface runoff (with the aid of an appropriate sub-model), infiltration, re turn flow, change in moisture storage and actual evapotranspiration (with the aid of an appropriate sub- model).

An empirical criterion for selection of the time step of simulation and a sub-model for specifying the h-O relation (i.e. h' soil moisture characteristic) are also provided for optimal use. A priori justification is presented in Mohan Rao et al. (1986). The model has the following assumptions: (i) stable and isotropic porous medium; (ii) immiscible two phase (air and water) and isothermal flow; (iii) negligible air phase flow with the air being at atmospheric~ pressure (arbitrarily taken as zero) throughout the unsaturated zone during simulation; (iv) water is pure and incompressible.

The present model

Assumptions Apart from the assumptions checklisted for the distributed model in the

previous section, the following additional assumptions are made. (1) No moisture movement at moisture contents less than or equal to field

capacity. (2) Complete drainage of excess moisture at moisture contents greater than

field capacity, in the basic accounting period.

234 K.M. MOHAN RAO ET AL.

(3) Uniform extraction of moisture by plant roots throughout the root zone and uniform distribution of moisture in the root zone, in the basic accounting. period.

(4) Soil medium is homogeneous. (5) The part of the unsaturated zone below the root zone will always be at

field capacity and so acts as a passive pathway for drainage of excess moisture. (6) The water table is deep enough so that the concept of field capacity is not

invalidated.

M e t h o d o f o p e r a t i o n (1) (Fj), the infiltration in the j th (basic accounting) period is added to (Sj),

the soil moisture storage at the start of the j th period. Thus, the revised uniform moisture (S]) will be:

Sj' = S~ + F~ (2a)

(2) The uniform actual evapotranspiration in the j th period (Ej) is calculated with the aid of an appropriate sub-model. The uniform moisture (S 7) after accounting for the actual evapotranspiration will be:

Sj" = S~ - Ej (2b)

(3) (SMj) the uniform maximum moisture that can be held by the soil in the root zone, in the j th period, is:

SIVI~ = F C . D t (2c)

where F C = field capacity and D t = depth of root zone in the j th period. (4) The return flow (Rt) is the excess of moisture above (SMj).

Thus:

Rj = S 7 - SM i i f S 7 > SM~ (2d)

= 0 if S/ ~< SM~ (2dd)

(5) If (Rt) evaluated at Step 4 is non-zero, the uniform moisture (S]") in the root zone equals (SM i). Otherwise, (S]") equals (S 7).

Sj" = SM~ if R t > 0 (2e)

= S 7 if R t = 0 (2ee)

(6) If the depth of root zone increases in the subsequent ( j + 1)st period, the uniform initial condition (St+,) for the ( j + 1)st period will be

St+ 1 = S]" + (Dr+ 1 - D t ) . F C (2f)

(7) If the depth of root zone decreases in the subsequent ( j + 1)st period, the uniform initial condition (S t+, ) for the ( j + 1)st period will be :

Sj+, = S ] " . D t + , / D t (2g)

SOIL MOISTURE ACCOUNTING MODEL 235

Field capacity as a flow parameter The concept of field capacity, in the SMA models, implies that soil moisture

movement occurs only if the moisture content exceeds the field capacity of the soil. Theoretically, this is possible if capillary conductivity (K), for moisture content (0), up to field capacity is zero as the gradients of flow (below field capacity) are not unconditionally zero. Further, the value of K beyond this point should be large enough to ensure complete drainage of excess moisture instantaneously (or virtually, in the basic accounting period). Thus, the K soil moisture characterist ic should be in accordance with eqn. (3):

K ( F C - ~) = 0 (3a)

K(FC + ~) = K, (35)

where:

= an infinitesimally small positive value K = capillary conductivity [LT-1) K, = saturated capillary conductivity (having a large value) [LT-1].

However, the plot of the K soil moisture characterist ic implied by eqn. (3) may deviate considerably from the usual plots (Fig. 1), particularly in the case of fine soils like clay. Thus, the concept of field capacity may be only approxim- ately true. Hence, the field capacity may be approximated by picking such a value of 0 below which the K characterist ic shows a rapid downward turn. For instance, points FC, (0 = 0.225), FCb (0 = 0.150) and FCc (0 -- 0.125) in Fig. 1 show field capacity marked for clay, silty clay loam and loam, respectively. The corresponding moisture levels at 0.33 bar tension as reported by Rawls et al. (1981) are 0.396, 0.366 and 0.270, respectively.

Data base

Crop activities The relevant crop activity details, as per the local practice (in and around

the eastern Yamuna Canal command area, India), are presented in Table 1. -2

10 LOAM 7

103 ~ LOAMI

/ / ', !

/ / I i i / l l / i ! , < , , i , , , , , , t

O0 0-05 O.tO FC c 0.15 0.20 FCo 0-25 0.30 0.35 0.40 0.45 0,475

Fig. 1. Real life 'K characteristic' with field capacity marked as per the proposed method.

236

TABLE 1

Crop activities

K.M. MOHAN RAO ET AL.

Crop Pre-sowing Sowing Harvesting (field) preparation

Rice 1-5 Jul. 6 Jul. 15 Oct. Wheat 15-19 Nov. 20 Nov. 15 Mar.

K soi l m o i s t u r e charac t e r i s t i c

The re la t ion (eqn. (4)) proposed by Brooks and Corey (1964) is adopted.

g = K ~ . [ ( O - 0r)/((~- 0r)] 4 0 /> 0r (4a)

= 0 0 < Or (4b)

where Or = res idual mois ture content , and ~b = soil porosity. The required numer ica l values for 0r, ¢ and K s are adopted from Rawls et al. (1981). These are: 0.09, 0.475, 1.667 E - 4 m m s -1, respectively, for the clay and 0.027, 0.463, 1.889 E - 3 mm s -1, respectively, for loam. Figure 1 shows the corresponding plots.

S u b - m o d e l f o r a c t u a l e v a p o t r a n s p i r a t i o n (E) Calcula ted month ly evapot ransp i ra t ion of the reference crop, near the

ra ingauge stat ion, is assumed to be uniform th roughou t the respective month. Using crop coefficients (based on crop and stage of growth) recommended by Doorenbos and Kassam (1979), potent ia l evapot ransp i ra t ion (Ep) is ca lcula ted for each crop every day. Table 2 presents the dai ly potent ia l evapotranspira- t ion of the reference crop, the crop coefficients and the dai ly potent ia l evapot ransp i ra t ion of the crops. For whea t and fallow crops, the Ep, E and 0 are re la ted as follows:

E/Ep = 0 if 0 <~ 0 ~ W P or ~b e < 0 ~< ~b (5a)

= ( 0 - W P ) / ( O t - W P ) if W P < 0 ~< 0t (5b)

= 1 if 0t < 0 ~< ~b e (5c)

where:

Ot = F C - p ( F C - W P ) : A dummy variable F C = field capaci ty W P = wil t ing point ( F C - W P ) = avai lable soil mois ture con ten t p = the f rac t ion of avai lable soil mois ture af ter which E = Ep ~be = (¢ - 0r) = effective soil porosity.

The p value for whea t in eqn. (5) is adopted from Doorenbos et al. (1979) and is presented in Table 2. In the case of fallow, p is assumed to be zero. The field

SOIL MOISTURE ACCOUNTING MODEL 237

R ~J

0

0 0~

g ~

R

0

~3

C~ 0 CO

v

0

O0

0 Z C3

0 c~ o0

oo

0

o~

q

0 qD

d R ~J

o

v

238 K,M. MOHAN RAO ET AL.

2eO0

260(

24-0C i

I 2200:

E 2000

z

1800

1600

~ 1400

1200 :>.

~ 800

6O0

4OO

SMA MODEL 1"

/

20 4.0 60 I]0 100 120 1'10 160 180 200 220 2.40 2.60 2BO 300 320 340 36S DAYS '

Fig. 2. Daily cumulative return flow by the SMA model and by the distributed model: rice-wheat cropping on clay.

capacities, as quantified from the K characteristic (Fig. 1) by the method proposed above are 0.225 and 0.125 for clay and loam, respectively. In the case of rice, since the input has been designed to maintain moisture content at or above field capacity (Mohan Rao et al., 1986), the actual evapotranspiration in the root zone is assumed to be at the potential rate.

Root zone depth At fortnightly intervals, data on root zone depths based on local observa-

tion, for various crops, are adopted. The depth is assumed to be constant over the fortnight and the change is assumed to take place instantaneously at the turn of the fortnight. Further, a minimum depth of 30 cm is adopted by assuming that direct evaporation from the soil takes place from the top 30 cm.

Relative performance of the SMA model

Figures 2 and 3 show the daily cumulative return flows obtained by the SMA model (with a basic accounting period of one day) and by the distributed model for the rice and wheat crops on clay and on loam, respectively. The slope of the curves provides the daily rates. The curves of the SMA model show pulsatory return flow, whereas those of the distributed model show continuous return

S O I L M O I S T U R E A C C O U N T I N G M O D E L 239

I1200 E E

z 1000

e 8oo

600

400

200

c~ j 0

, f

20 40 60 80 100 120 140

DAY

OIST. MODEL SMA,~MODEL/ - - ~ ~

I I I I I l I I I I 160 100 200 220 240 260 200 300 320 340 3 6 5

Fig. 3. Daily cumulative return flow by the SMA model and by the distributed model: rice-wheat cropping on loam.

flow. The daily rates predicted by the two models differ considerably. Further, the SMA model tends to overestimate the return flow rates during the early period of peak infiltration (rainy season with the rice crop), because the SMA model does not account for the time lag in the occurrence of return flow. Moreover, rice is a shallow-rooted crop and hence the moisture deficiency to be replenished is less before the return flow occurs. As infiltration decreases, subsequently, (winter season with the wheat crop), the SMA model predicts lower rates of return flow, particularly in the case of the clay soil. Thus, the seasonal totals of return flow tend to match those predicted by the distributed model (Table 3).

C O N C L U S I O N S

Soil moisture accounting models involve many restrictive assumptions, and as a result the models lead to discrepancies in time distribution of estimates of return flow. T h e assumption of the concept of the field capacity, often quantified at 0.33 bar tension, may be made a more realistic flow parameter by

TABLE 3

Comparison of seasonal totals of return flows

Soil cropping Return flow (mm) by the end of

Rice season Wheat season One year

Rice-wheat on clay S M A m o d e l 2625.32

Dist. Model 1730.66

Rice-wheat on loam SMAmode l 395.65

Dist. Model 176.61

2675.66 2675.66

2403.16 2526.59

470.65 470.65

491.47 571.83

240 K.M. MOHAN RAO ET AL.

quan t i fy ing it f rom the cap i l l a ry c o n d u c t i v i t y - m o i s t u r e con t en t re la t ion . One such method , which is more su i t ab le for coa r se soils, is p roposed in the p re sen t w o r k and i m p l e m e n t e d in the p roposed SMA model.

D i s t r ibu ted models (based on so lu t ion of the R icha rds ' equa t ion) a re free f rom m a n y of the r e s t r i c t i ve a s sumpt ions a s soc ia t ed wi th the SMA models , p a r t i c u l a r l y t h a t of field capac i ty . Thus, c o m p a r i s o n of the r e t u r n flow es t ima tes of bo th these models , would show the impac t of the r e s t r i c t ive a s s u m p t i o n s in the SMA models . Use of an iden t ica l da t a base showed r e a s o n a b l e a g r e e m e n t be tween the seasona l to t a l s of r e t u r n flow as p red ic ted by bo th models. However , the dai ly r a t e s differ cons iderably . These observa- t ions show the effects of not a c c o u n t i n g for the t ime lag on occur rence , un- d e r e s t i m a t i o n and o v e r e s t i m a t i o n of the r e t u r n flow by the SMA model. Fu r the r , in genera l , the p e r f o r m a n c e of the SMA model is more accep tab le (in v iew of the ob jec t ives of the work) in the case of the loam soil (coarse soil), p e rhaps because the quan t i f i ca t ion of field c apac i t y as a flow p a r a m e t e r by the p roposed me thod is more su i t ab le for coarse soils.

ACKNOWLEDGEMENT

The p resen t work is a pa r t of the r e s ea r ch p ro jec t en t i t l ed ' C o n j u n c t i v e Use Model l ing of G r o u n d and Sur face Wa t e r s in E a s t e r n Y a m u n a Cana l C o m m a n d Area ' , conduc ted by the first two au thors , which was sponsored by the Cen t ra l Board of I r r i g a t i o n and P o w e r ( G o v e r n m e n t of India) . The a u t h o r s wish to express the i r t h a n k s to the Board.

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SOIL MOISTURE ACCOUNTING MODEL 241

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Sykes, D.J. and Loomis, W.E, 1967. Plant and soil factors in permanent wilting percentages and field capacity storage. Soil Sci., 104: 163-173.

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